decompwlj 3D

Numbers that are the sum of 2 distinct nonzero squares

A004431 on the OEIS · family quadratic form · also known as Sums of two distinct nonzero squares

Weight–level plate of Numbers that are the sum of 2 distinct nonzero squares
Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.
Open in the 3-D viewerA004431 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L15,977 · 15.98 %
Weight class, k ≤ L84,022 · 84.02 %
Ties, k = L15
On the level line L = 16,152
Forced level, l ≤ d²4
Range of a(n)5 … 449,105
Range of the jump d1 … 34
Largest weight k, level L449,051, 224,544

x^2 + y^2 with 0 < x < y. The plane is that of all sums of two squares (15.98 % level against 15.95 %). The level share is 15.98 %; L = 1 holds 39 % of the level class.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.